Disjoining Pressures

FIGURE 1.8 The liquid profile in the vicinity of the apparent three-phase contact hne (1) bulk liquid, where boundary layers do not overlap, (2) boundary layer in the vicinity liquid-air and liquid-solid interfaces, (3) a region where boundary layers overlap, and (4) flat thin equilibrium film. The latter two are the regions where disjoining pressure acts. 

The main conclusion the pressure in thin layers close to the three-phase contact line is different from the pressure in the bulk liquid, and it depends on the thickness of the layer, h, and varies with the thickness, h. 

In the following, we briefly review the physical phenomena that result in the formation of the above mentioned surface forces and disjoining pressure. 

Several physical phenomena have been identified for the appearance of the disjoining pressure. Here, we consider only three of them. 

---

Thus it is necessary to find alternative approach to describe the physical mechanism of two-side filling of conical capillaries with hquids. Theoretical model of film flow in conical dead-end capillary is based on the concept of disjoining pressure II in thin liquid film [13]... 

Use well-known approximation for the isotherm of disjoining pressure ri(h) = AJh, where A is the constant. The film s curvature can be expressed as K = -l/(tr). After some transformations we derive the equation for the thickness h of liquid film ... 

Fig. 4 illustrates the time-dependence of the length of top s water column in conical capillary of the dimensions R = 15 pm and lo =310 pm at temperature T = 22°C. Experimental data for the top s column are approximated by the formula (11). The value of A is selected under the requirement to ensure optimum correlation between experimental and theoretical data. It gives Ae =3,810 J. One can see that there is satisfactory correlation between experimental and theoretical dependencies. Moreover, the value Ae has the same order of magnitude as Hamaker constant Ah. But just Ah describes one of the main components of disjoining pressure IT [13]. It confirms the rightness of our physical arguments, described above, to explain the mechanism of two-side liquid penetration into dead-end capillaries. 

One more experimental result, which is important for PT is as follows. Only polar liquids fill conical capillaries from both sides. We used various penetrants to fill conical defects Pion , LZh-6A , LZhT , LUM-9 etc. It was established that only the penetrants containing polar liquid as the basic liquid component (various alcohols, water and others) manifest two-side filling phenomenon. This result gives one more confirmation of the physical mechanism of the phenomenon, based on liquid film flow, because the disjoining pressure strongly depends just on the polarity of a liquid. 

In the preceding derivation, the repulsion between overlapping double layers has been described by an increase in the osmotic pressure between the two planes. A closely related but more general concept of the disjoining pressure was introduced by Deijaguin [30]. This is defined as the difference between the thermodynamic equilibrium state pressure applied to surfaces separated by a film and the pressure in the bulk phase with which the film is equilibrated (see section VI-5). 

Consider the situation illustrated in Fig. VI-9, in which two air bubbles, formed in a liquid, are pressed against each other so that a liquid film is present between them. Relate the disjoining pressure of the film to the Laplace pressure P in the air bubbles. 

Deijaguin and Zorin report that at 25°C, water at 0.98 of the saturation vapor pressure adsorbs on quartz to give a film 40 A thick. Calculate the value of the disjoining pressure of this film and give its sign. 

Equation X-43 may, alternatively, be given in terms of a disjoining pressure (Section VI-6) integrd [187] ... 

Fig. 2. Effective interface potential (left) and corresponding disjoining pressure (right) vs film thickness as predicted by DLVO theory for an aqueous soap film containing 1 mM of 1 1 electrolyte. The local minimum in H(f), marked by °, gives the equiHbrium film thickness in the absence of appHed pressure as 130 nm the disjoining pressure 11 = —(dV/di vanishes at this minimum. The minimum is extremely shallow compared with the stabilizing energy barrier.

Disjoining Pressure. A static pressure difference can be imposed between the interior and exterior of a soap film by several means including, for example, gravity. In such cases the equiHbrium film thickness depends on the imposed pressure difference as weU as on the effective interface potential. When the film thickness does not minimize lV(f), there arises a disjoining pressure II = —dV/(U which drives the system towards mechanical equiHbrium. 

In response to a hydrostatic pressure, the film thickness thus adjusts itself so that the disjoining pressure balances the appHed pressure and mechanical equiHbrium is restored. 

Theoretically, the diffusion coefficient can be described as a function of the disjoining pressure 77, the effective viscosity of lubricant, 77, and the friction between lubricant and solid surfaces. In relatively thick films, an expression derived from hydrodynamics applies to the diffusion coefficients. 

The dependence on film thickness is attributed to the dewetting nucleation, which occurs in the 2.5-4.5 nm thickness range via the formation of randomly distributed droplets rather than the formation of holes. When the initial film thickness exceeds 4.5 nm, dewetting is trigged via nucleation of holes instead of droplets, and for film thickness above 10 nm, dewetting develops slowly via hole nucleation at defects. The different dewetting processes observed for different initial film thicknesses can be explained in terms of the variation of disjoining pressure and the inability of the polymer to spread on its own monolayer. 

A quantity called disjoining pressure II was introduced by Derjaguin [2,3] as the natural canonical conjngate of the film thickness, e ... 

The disjoining pressure characterizes the wetting properties at short ranges. S and II are related by ... 

In a real liqnid, the nature of the disjoining pressure can be a complicated function of the distance, due to the simultaneous coutributiou of several types of forces [1,4-6], For two bodies with flat surfaces separated by a distance z, the van der Waals interaction, which varies as z (or z, if one considers retardation) for single atoms and molecules, gives rise to a power law of the form ... 

Althongh van der Waals forces are present in every system, they dominate the disjoining pressnre in only a few simple cases, such as interactions of nonpolar and inert atoms and molecnles. It is common for surfaces to be charged, particularly when exposed to water or a liquid with a high dielectric constant, due to the dissociation of surface ionic groups or adsorption of ions from solution, hi these cases, repulsive double-layer forces originating from electrostatic and entropic interactions may dominate the disjoining pressure. These forces decay exponentially [5,6] ... 

From Eq. (9), the following relation between the effective contact angle and the disjoining pressure under constant volume can be obtained ... 

Capillary phenomena are due to the curvature of liquid surfaces. To maintain a curved surface, a force is needed. In Eq. (8), this force is related to the second term in the integral. The so-called Laplace pressure due to the force to maintain the curved surface can be expressed as, Pl = 2-yIr, where r is the curvature radius. The combination of the capillary effects and disjoining pressure can make a liquid film climb a wall. 

A. Disjoining Pressure Effects on the Contact Angle of Small Droplets... 

FIG. 7 Effective contact angle of the aqueous KOH droplets on HOPG and mica as a function of droplet height. Solid lines correspond to fits obtained using the disjoining pressure given by Eq. (18). 

In this case, the hydrophobic interaction is very weak compared to that of aq.KOH-graphite system. In spite of this, it still dominates the disjoining pressure. 

---